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A box contains an inductor of inductance L, a capacitor of capacitance C and a resistor of resistance R. When 250 V d. c. is applied to the terminals of the box, a current of 1.0 A plus an oscillating current flows in the circuit. When an a. c. source of 250 V (r.m.s) at 2250 rad/sec is connected, an rms current of 1.25 A is produced. This current rises with frequency and becomes maximum at 4500 rad/sec. Evaluate $2 R \sqrt{C / L}$

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answer is 2.25.

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Detailed Solution

The constant + oscillating current shows that the L and C are in series, and this combination is in parallel with R = 250 ohm.

$\frac{1}{Z} = \frac{1}{Z_{1}} + \frac{1}{Z_{2}}$

$\frac{1.25}{250} = | \frac{1}{250} + \frac{1}{i ω L + \frac{1}{i ω C}} |$

$| \frac{1}{250} + \frac{i ω C}{1 + ω^{2} L C} |$

$| \frac{1}{250} + \frac{i 2250 C}{1 - 0.25} |$

Hence: $\frac{1}{200^{2}} = \frac{1}{250^{2}} + \frac{16}{9} \times 2250^{2} C^{2}$

Hence $C = 10^{- 6} F .$ Also $4500^{2} L C = 1$

Hence $L = \frac{4}{81} H$

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